# Hooke’s Law

Hooke’s Law states that, in a linear system, the restoring force is proportional to the displacement of the body, acting in a direction as to restore equilibrium.

F = -kx

where:
F = restoring force [N]

k = spring constant [Nm-1]

x = elongation of material [m]

Robert Hooke discovered the law of elasticity which bears his name in 1660. This describes the linear variation of tension with extension in an elastic spring. He first announced his law of elasticity as an anagram to to

Robert Hooke, De Potentia Restitutiva, or of Spring. Explaining the Power of Springing Bodies, London, 1678.

When we plot force versus extension of a material we see the linear region and then the non-linear behaviour. Finally we see the failure of the material.

A material such as copper, when stretched beyond it’s elastic limit it retains it’s shape.

A material such as rubber, this does not follow Hooke’s law and remains elastic until it snaps.

Just before failure you see the force increasing for a given unit of extension.

A material such as glass follows Hooke’s law until it snaps.

#### Apparent Elastic Limit

Arbitrary approximation of the elastic limit of materials that do not have a significant straight line portion on a stress strain diagram.

#### Elastic Deformation

A non-permanent deformation that totally recovers upon release of an applied stress.

#### Elastic Strain

Dimensional changes accompanying stress where the original dimensions are restored upon release of the stress.

#### Elasticity

A material is elastic if it returns to its original shape after being deformed. The maximum load that a body can experience and still return to its original shape is known as the Elastic Limit.

• Metals: The elastic limit is defined as the 0.2% offset yield strength. This represents the stress at which the stress-strain curve for uniaxial tensile loading deviates by a strain of 0.2% from the linear-elastic line. It is the same in tension and compression. It is the stress at which dislocations move a large distance through the crystals of the metal.
• Polymers: The elastic limit is the stress at which the uniaxial stress-strain curve becomes markedly non-linear: typically, a strain of 1%. This may be caused by ‘shear yielding’ (irreversible slipping of molecular chains) or by ‘crazing’ (formation of low density, crack-like volumes which scatter light, making the polymer look white).
• Fine Ceramics and Glasses: The elastic limit is often an estimate, based on the tensile strength (which is low due to brittle fracture).
• Composites: The elastic limit is best defined by a set deviation from linear-elastic uniaxial behaviour.

#### Offset Yield Strength

Arbitrary approximation of elastic limit.

It is the stress that corresponds to the point of intersection of a stress-strain diagram and a line parallel to the straight line portion of the diagram.

Offset refers to the distance between the origin of the stress-strain diagram, and the point of intersection of the parallel line and the 0 stress axis.

Offset is expressed in terms of strain (often 0.2%).

#### Plasticity

The inverse of elasticity. A material that tends to stay in the shape or size to which it is deformed has high plasticity.

#### Spring Constant

The amount of force required to stretch a spring by 1 unit of length. Measured in newtons per meter.

#### Youngs Modulus

The constant of that proportionality is the Young modulus of elasticity for that substance.

E = Young’s modulus [Nm-2]

F = applied load [N] A = cross-sectional area [m2]

x = extension [m]

L = original length [m]

Stress – strain curve for a material such as mild steel.

Hooke’s Law would apply within the linear region up to the elastic limit of the material.